# §10.49 Explicit Formulas

## §10.49(i) Unmodified Functions

Define $a_{k}(\nu)$ as in (10.17.1). Then

 10.49.1 $a_{k}(n+\tfrac{1}{2})=\begin{cases}\dfrac{(n+k)!}{2^{k}k!(n-k)!},&k=0,1,\ldots% ,n,\\ 0,&k=n+1,n+2,\ldots.\end{cases}$ ⓘ Symbols: $!$: factorial (as in $n!$), $n$: integer, $k$: nonnegative integer and $a_{k}(\nu)$: expansion Referenced by: §10.49(ii), §10.50 Permalink: http://dlmf.nist.gov/10.49.E1 Encodings: TeX, pMML, png See also: Annotations for 10.49(i), 10.49 and 10
 10.49.2 $\mathsf{j}_{n}\left(z\right)=\sin\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{% \left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+% \cos\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor(n-1)/2\right% \rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}.$
 10.49.3 $\displaystyle\mathsf{j}_{0}\left(z\right)$ $\displaystyle=\frac{\sin z}{z},$ $\displaystyle\mathsf{j}_{1}\left(z\right)$ $\displaystyle=\frac{\sin z}{z^{2}}-\frac{\cos z}{z},$ $\displaystyle\mathsf{j}_{2}\left(z\right)$ $\displaystyle=\left(-\frac{1}{z}+\frac{3}{z^{3}}\right)\sin z-\frac{3}{z^{2}}% \cos z.$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$: spherical Bessel function of the first kind and $z$: complex variable A&S Ref: 10.1.11 (corrected) Referenced by: §10.49(iii), §10.56 Permalink: http://dlmf.nist.gov/10.49.E3 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 10.49(i), 10.49 and 10
 10.49.4 $\mathsf{y}_{n}\left(z\right)=-\cos\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{% \left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+% \sin\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor(n-1)/2\right% \rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}.$
 10.49.5 $\displaystyle\mathsf{y}_{0}\left(z\right)$ $\displaystyle=-\frac{\cos z}{z},$ $\displaystyle\mathsf{y}_{1}\left(z\right)$ $\displaystyle=-\frac{\cos z}{z^{2}}-\frac{\sin z}{z},$ $\displaystyle\mathsf{y}_{2}\left(z\right)$ $\displaystyle=\left(\frac{1}{z}-\frac{3}{z^{3}}\right)\cos z-\frac{3}{z^{2}}% \sin z.$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$: spherical Bessel function of the second kind and $z$: complex variable A&S Ref: 10.1.12 (corrected) Referenced by: §10.50, §10.56 Permalink: http://dlmf.nist.gov/10.49.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 10.49(i), 10.49 and 10
 10.49.6 $\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)$ $\displaystyle=e^{iz}\sum_{k=0}^{n}i^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}},$ 10.49.7 $\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)$ $\displaystyle=e^{-iz}\sum_{k=0}^{n}(-i)^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{% k+1}}.$

## §10.49(ii) Modified Functions

Again, with $a_{k}(n+\tfrac{1}{2})$ as in (10.49.1),

 10.49.8 ${\mathsf{i}^{(1)}_{n}}\left(z\right)=\tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}% \frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n+1}\*\tfrac{1}{2}e^{-z}\sum_{k=0}^% {n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}.$ ⓘ Symbols: $\mathrm{e}$: base of exponential function, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_{k}(\nu)$: expansion A&S Ref: 10.2.9 (modified) Referenced by: §10.52(ii) Permalink: http://dlmf.nist.gov/10.49.E8 Encodings: TeX, pMML, png See also: Annotations for 10.49(ii), 10.49 and 10
 10.49.9 $\displaystyle{\mathsf{i}^{(1)}_{0}}\left(z\right)$ $\displaystyle=\frac{\sinh z}{z},$ $\displaystyle{\mathsf{i}^{(1)}_{1}}\left(z\right)$ $\displaystyle=-\frac{\sinh z}{z^{2}}+\frac{\cosh z}{z},$ $\displaystyle{\mathsf{i}^{(1)}_{2}}\left(z\right)$ $\displaystyle=\left(\frac{1}{z}+\frac{3}{z^{3}}\right)\sinh z-\frac{3}{z^{2}}% \cosh z.$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$: modified spherical Bessel function and $z$: complex variable A&S Ref: 10.2.13 Permalink: http://dlmf.nist.gov/10.49.E9 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 10.49(ii), 10.49 and 10
 10.49.10 ${\mathsf{i}^{(2)}_{n}}\left(z\right)=\tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}% \frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n}\tfrac{1}{2}e^{-z}\sum_{k=0}^{n}% \frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}.$ ⓘ Symbols: $\mathrm{e}$: base of exponential function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_{k}(\nu)$: expansion A&S Ref: 10.2.10 (modified) Referenced by: §10.52(ii) Permalink: http://dlmf.nist.gov/10.49.E10 Encodings: TeX, pMML, png See also: Annotations for 10.49(ii), 10.49 and 10
 10.49.11 $\displaystyle{\mathsf{i}^{(2)}_{0}}\left(z\right)$ $\displaystyle=\frac{\cosh z}{z},$ $\displaystyle{\mathsf{i}^{(2)}_{1}}\left(z\right)$ $\displaystyle=-\frac{\cosh z}{z^{2}}+\frac{\sinh z}{z},$ $\displaystyle{\mathsf{i}^{(2)}_{2}}\left(z\right)$ $\displaystyle=\left(\frac{1}{z}+\frac{3}{z^{3}}\right)\cosh z-\frac{3}{z^{2}}% \sinh z.$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$: modified spherical Bessel function and $z$: complex variable A&S Ref: 10.2.14 Permalink: http://dlmf.nist.gov/10.49.E11 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 10.49(ii), 10.49 and 10
 10.49.12 $\mathsf{k}_{n}\left(z\right)=\tfrac{1}{2}\pi e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n% +\frac{1}{2})}{z^{k+1}}.$
 10.49.13 $\displaystyle\mathsf{k}_{0}\left(z\right)$ $\displaystyle=\tfrac{1}{2}\pi\frac{e^{-z}}{z},$ $\displaystyle\mathsf{k}_{1}\left(z\right)$ $\displaystyle=\tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{1}{z^{2}}\right),$ $\displaystyle\mathsf{k}_{2}\left(z\right)$ $\displaystyle=\tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{3}{z^{2}}+\frac{3}% {z^{3}}\right).$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$: modified spherical Bessel function and $z$: complex variable A&S Ref: 10.2.17 Permalink: http://dlmf.nist.gov/10.49.E13 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 10.49(ii), 10.49 and 10

$\sum_{k=0}^{n}a_{k}(n+\tfrac{1}{2})z^{n-k}$ is sometimes called the Bessel polynomial of degree $n$. For a survey of properties of these polynomials and their generalizations see Grosswald (1978). See also §18.34, de Bruin et al. (1981a, b), and Dunster (2001c).

## §10.49(iii) Rayleigh’s Formulas

 10.49.14 $\displaystyle\mathsf{j}_{n}\left(z\right)$ $\displaystyle=z^{n}\left(-\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}% \frac{\sin z}{z},$ $\displaystyle\mathsf{y}_{n}\left(z\right)$ $\displaystyle=-z^{n}\left(-\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n% }\frac{\cos z}{z}.$
 10.49.15 $\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)$ $\displaystyle=z^{n}\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}% \frac{\sinh z}{z},$ $\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)$ $\displaystyle=z^{n}\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}% \frac{\cosh z}{z}.$
 10.49.16 $\mathsf{k}_{n}\left(z\right)=(-1)^{n}\tfrac{1}{2}\pi z^{n}\left(\frac{1}{z}% \frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}\frac{e^{-z}}{z}.$

## §10.49(iv) Sums or Differences of Squares

Denote

 10.49.17 $s_{k}(n+\tfrac{1}{2})=\frac{(2k)!(n+k)!}{2^{2k}(k!)^{2}(n-k)!},$ $k=0,1,\ldots,n$. ⓘ Defines: $s_{k}(n)$ (locally) Symbols: $!$: factorial (as in $n!$), $n$: integer and $k$: nonnegative integer Permalink: http://dlmf.nist.gov/10.49.E17 Encodings: TeX, pMML, png See also: Annotations for 10.49(iv), 10.49 and 10

Then

 10.49.18 ${\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}^{2}}\left(z\right)=\sum_{k=% 0}^{n}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}.$
 10.49.19 $\displaystyle{\mathsf{j}_{0}^{2}}\left(z\right)+{\mathsf{y}_{0}^{2}}\left(z\right)$ $\displaystyle=z^{-2},$ $\displaystyle{\mathsf{j}_{1}^{2}}\left(z\right)+{\mathsf{y}_{1}^{2}}\left(z\right)$ $\displaystyle=z^{-2}+z^{-4},$ $\displaystyle{\mathsf{j}_{2}^{2}}\left(z\right)+{\mathsf{y}_{2}^{2}}\left(z\right)$ $\displaystyle=z^{-2}+3z^{-4}+9z^{-6}.$ ⓘ Symbols: $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$: spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$: spherical Bessel function of the second kind and $z$: complex variable A&S Ref: 10.1.28--10.1.30 Permalink: http://dlmf.nist.gov/10.49.E19 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 10.49(iv), 10.49 and 10
 10.49.20 $\left({\mathsf{i}^{(1)}_{n}}\left(z\right)\right)^{2}-\left({\mathsf{i}^{(2)}_% {n}}\left(z\right)\right)^{2}=(-1)^{n+1}\sum_{k=0}^{n}(-1)^{k}\frac{s_{k}(n+% \frac{1}{2})}{z^{2k+2}}.$
 10.49.21 $\displaystyle\Big{(}{\mathsf{i}^{(1)}_{0}}\left(z\right)\Big{)}^{2}-\Big{(}{% \mathsf{i}^{(2)}_{0}}\left(z\right)\Big{)}^{2}$ $\displaystyle=-z^{-2},$ $\displaystyle\Big{(}{\mathsf{i}^{(1)}_{1}}\left(z\right)\Big{)}^{2}-\Big{(}{% \mathsf{i}^{(2)}_{1}}\left(z\right)\Big{)}^{2}$ $\displaystyle=z^{-2}-z^{-4},$ $\displaystyle\Big{(}{\mathsf{i}^{(1)}_{2}}\left(z\right)\Big{)}^{2}-\Big{(}{% \mathsf{i}^{(2)}_{2}}\left(z\right)\Big{)}^{2}$ $\displaystyle=-z^{-2}+3z^{-4}-9z^{-6}.$ ⓘ Symbols: ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$: modified spherical Bessel function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$: modified spherical Bessel function and $z$: complex variable A&S Ref: 10.2.27--10.2.29 Permalink: http://dlmf.nist.gov/10.49.E21 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 10.49(iv), 10.49 and 10